Assuming they can be effectively programmed, parallel processors such as graphics processing units (GPUs) have the potential to be remarkably adept at processing numerical algorithms, and particularly algorithms for directly solving large sparse linear systems.
Sparse linear systems are systems of linear equations with sparse coefficient matrices. These systems arise in the context of computational mechanics, geophysics, biology, circuit simulation and many other contexts in the fields of computational science and engineering.
The most common general and direct technique to solve a sparse linear system is to decompose its coefficient matrix into the product of a lower triangular matrix, L, and an upper triangular matrix, U, a process called “factorization.” Then, conventional forward and backward substitution techniques can be used to solve the linear systems with L and U triangular matrices and thereby obtain the solution of the sparse linear system.